4.2 Simulation Methods
4.2.2 Structural Analysis
Bond Order Parameters
The structure of the clusters is studied using a number of order parameters based on the Steinhardt[14] bond order parameters. The bond orientational order parameter, Ql, is defined as Ql = v u u t 4π 2l + 1 l X m=−l |1 N N X i=1 qlm(i)|2, (4.3) where qlm(i) = 1 Nnb Nnb(i) X j=1 Ylm(rij). (4.4)
The summation in eqn 4.4 is over the number of neighbors, (Nnb), for atom i, where two atoms are considered neighbors if the distance between them is less than or equal to 3.5 ˚A, which is the radius corresponding to the first minimum of the radial distribution function
200 400 600 800 1000 1200 1400 t(ps) -3.66 -3.64 -3.62 -3.6 -3.58 U/N(eV)
Figure 4.1: The energy of the cluster per atom as a function of simulation time for six trajectories (N = 561). The red solid line, U = −3.63eV , identifies the energy used to determine when a cluster has nucleated.
for gold. Ylm(rij) = Ylm(θij, φij) are the spherical harmonic functions, where θ and φ are the polar and azimuthal angles of the vector rij, respectively; l is is a free integer parameter, and m is an integer that runs from m = −l to m = +l. The value of qlm depends on the relative positions and orientations of the neighbors of atom i, and therefore allows the evaluation of the structures of clusters with different symmetries depending on the choice of l. For this work, l = 6 is chosen since Q6 is sensitive to hexagonal structures and gives non-zero Q6 values for the cluster structures observed in our simulations. Ql have been used extensively in studies of clusters [44, 131, 2] and liquid-solid nucleation in the bulk [132].
The Qs and Qb, which are just the Q6 order parameters calculated independently for the surface and bulk atoms of the cluster [2], are also measured. These help us to understand how surface ordering may play a role in the freezing of the clusters as well as distinguishing between different structure types. They are defined as
Qb,s= v u u t4π 13 6 X m=−6 |N1 b,s Nb,s X i=1 q6m(i)|2, (4.5)
surface atoms from the bulk atoms, a slightly modified “cone” algorithm[131] is used. For a given atom, a “cone region” is defined as the region inside a cone of side length, lc, with azimuthal angle, θc, and whose vertex rests on the atom center. An atom is said to be on the surface if it is possible to find a cone surrounding the particle that contains no other atoms, otherwise, the atom is regarded as bulk atom. Fig. 4.2 shows a diagram of how the cone algorithm is used to probe surface and bulk atoms. For this work, θc = 120◦ and lc = 3.5 ˚A.
!c lc !c lc ! ! !
Figure 4.2: A diagram showing how surface and bulk atoms are identified using the cone having azimuthal angle, θc, and side length, lc. The upper cone identifies a surface atom while the lower cone
shows a bulk atom.
Common Neighbor Analysis
The local order of the individual atoms is studied using a common neighbor analysis (CNA) [133, 134, 135]. This method, which identifies atoms by considering the number and connectivity of the neighbors shared by two neighboring atoms, was first proposed by Honeycutt and Anderson[136]. Considering a pair of atoms, α and β, the CNA is classified by a set of indexes, (i), (ii), (iii) and (iv). The indexes: (i) indicates whether α and β are nearest neigh- bors (i = 1) or not (i = 2); (ii) indicates the number of nearest neighbors shared by α and β, which are called common neighbors; (iii) indicates the number of bonds or connections among the common neighbors; (iv) indicates the number of bonds in the longest continuous chain formed by the common neighbors. In this work, two atoms are considered neighbors if the distance between them is less than or equal to 3.5 ˚A.
Fig. 4.3 is an illustration of a diagram constructed from the classification of local structural environments. Atoms i (brown) and j (yellow) are a pair of nearest neighbors, while atoms k (blue) are the common neighbors to the pair, i,j. From this figure it can be seen that i and j have 4 common neighbors, and these common neighbors have 2 bonds between themselves. Also, the pair bond with their common neighbors in the same way, with two bonds in the longest chain. Therefore, the CNA environment of i with respect to j is 1422. The local environment of an atom is determined in the both bulk and surface based on the number of CNA environments that an atom forms with its nearest neighbors. An atom is regarded as a bulk atom if the number of nearest neighbors is greater than or equal to 10, and a surface atom otherwise. For a bulk atom, if the number of pairs with index 1421 is equal or greater than 5, then that atom has an f cc local structure. It be noted that f cc is used to identify the local environment of the atom, while F CC identifies the entire cluster structure. Other indexes used for bulk atoms include 1422 and 1555. If the number of pairs for a given atom with 1555 is greater than or equal to 2, then the atom is bulk Ih, if the number of pairs with index 1422 is greater than or equal to 5, then the atom has hcp local environment. Bulk atoms not identified as f cc, Ih or hcp are regarded as amorphous. For surface atoms, if the number of pairs for a given atom with 1555 is greater than or equal to 1, then the atom is Ih − vertex. If the number of pairs for a given atom with 1211 is greater than or equal to 3, then the atom is on the 100 surface. There are some local environments on the surface, which cannot be identified by counting a single index. Examples of such local structures include, 111, Ih − edge, Ih − join and fcc − edge, therefore a combination of indices such as 1211,1311, 1322, and 1422 are used to identify such local environments. A surface atom not identified as being a vertex, on an edge, joint, on the 111 or 100 planes is regarded as an amorphous atom on the surface.
Finally, the effectiveness of our structural analysis is dramatically improved when per- formed on configurations that have been subjected to a conjugate gradient quench that takes the configuration to its local potential energy minimum, or an inherent structure [137], to remove the thermal noise from the structure. All the structural quantities reported in this chapter refer to our analysis on these quenched configurations.
Figure 4.3: Diagrams showing how the classification of local structures defined in CNA are con- structed. Reproduced with permission from ref. [135]